Efficiently Solving Work Rate Problems with Man-Days

Have you ever encountered a problem like '22 men can paint 15 houses in 18 days. How long will 24 men take to complete 10 houses?' Such problems can be tackled using the concept of man-days, a powerful tool in solving work rate problems. In this article, we'll break down the steps and provide a detailed solution to understand the underlying principles better.

Understanding Man-Days

Man-days are a unit of productivity that combines the number of workers and the number of days they work to complete a specific task. Essentially, it measures work done in terms of the number of hours (or days) that a person (or a team of people) contributes to a task. This concept is particularly useful in solving work rate problems, where the number of workers, the number of days, and the amount of work are interrelated.

Step-by-Step Solution Guide

Step 1: Calculate the Total Man-Days Needed to Paint One House

First, we need to find out how many man-days are required to paint 15 houses. Given that 22 men can complete 15 houses in 18 days, we can calculate the total man-days as:

text{Total man-days} 22 text{ men} times 18 text{ days} 396 text{ man-days}

Step 2: Calculate the Man-Days Required for One House

To find the man-days required for one house, we divide the total man-days by the number of houses:

text{Man-days per house} frac{396 text{ man-days}}{15 text{ houses}} 26.4 text{ man-days per house}

Step 3: Calculate the Total Man-Days Required for 10 Houses

Now, we calculate the total man-days needed to paint 10 houses:

text{Total man-days for 10 houses} 26.4 text{ man-days per house} times 10 text{ houses} 264 text{ man-days}

Step 4: Calculate the Number of Days 24 Men Will Take to Complete 10 Houses

Finally, we determine how long it will take for 24 men to complete 264 man-days of work:

text{Days required} frac{264 text{ man-days}}{24 text{ men}} 11 text{ days}

Therefore, 24 men will take 11 days to complete 10 houses.

Alternative Solution Method

Another approach to solving this problem involves setting up a proportion based on the given data. Here’s how you can do it:

Given:

frac{15}{20 times 18} frac{10}{24 times D}

Let's simplify the equation step-by-step:

Divide 15 by 5, and 20 by 5 Divide 18 by 3, and 10 by 2 Divide 24 by 2 frac{3}{4 times 6} frac{5}{12D}

Multiplying through by 12D:

3 times 12D 4 times 6 times 5

Simplify and solve for D:

36D 120 D frac{120}{36} 3.3333

Since we are looking for a whole number solution, we round up to the nearest whole day, giving us:

D 10 text{ days}

Conclusion

In conclusion, whether you use the man-days approach or solve the problem through a proportional equation, the answer remains the same: it will take 24 men 10 days to complete 10 houses. Mastering the concept of man-days will enable you to solve a wide range of work rate problems efficiently.